compute closed forms of indefinite sums of rational functions - Maple Help

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SumTools[IndefiniteSum][Rational] - compute closed forms of indefinite sums of rational functions

Calling Sequence

Rational(f, k, options)

Parameters

f

-

rational function in k

k

-

name

options

-

(optional) equation of the form failpoints=true or failpoints=false

Description

• 

The Rational(f, k) command computes a closed form of the indefinite sum of f with respect to k.

• 

Rational functions are summed using Abramov's algorithm (see the References section). For the input rational function fk, the algorithm computes two rational functions sk and tk such that fk=sk+1sk+tk and the denominator of tk has minimal degree with respect to k.  The non-rational part, ktk, is then expressed in terms of the digamma and polygamma functions.

• 

If the option failpoints=true (or just failpoints for short) is specified, then the command returns a pair g,p,q, where

– 

g is the closed form of the indefinite sum of f w.r.t. k,

– 

p is a list containing the integer poles of f, and

– 

q is a list containing the poles of s and t that are not poles of f.

  

See SumTools[IndefiniteSum][Indefinite] for more detailed help.

Examples

withSumTools[IndefiniteSum]:

The following expression is rationally summable.

f:=1n2+5n1

f:=1n2+5n1

(1)

g:=Rationalf,n

g:=13n32+12513n12+12513n+12+125

(2)

Check the telescoping equation:

evalaNormalgn=n+1|gn=n+1g,expanded

1n2+5n1

(3)

A non-rationally summable example.

f:=1357x+2y+20x218xy+10y215+10x26y25x2+10xy+8y2

f:=20x218xy+10y257x+2y+1325x2+10xy+8y2+10x26y+15

(4)

g:=Rationalf,x

g:=45x+725y+3425Ψx45y+35+1725y+35Ψx+25y1

(5)

simplifycombinefgx=x+1|gx=x+1g,Ψ

0

(6)

Compute the fail points.

f:=1n2n3+1n5

f:=1n2n3+1n5

(7)

g,fp:=Rationalf,n,'failpoints'

g,fp:=1n51n4+1n3+1n2+1n1,0..0,3..3,5..5,1,2,4

(8)

Indeed, f is not defined at n=0,3,5, and g is not defined at n=1,2,4.

See Also

SumTools[IndefiniteSum], SumTools[IndefiniteSum][Indefinite]

References

• 

Abramov, S.A. "Indefinite sums of rational functions." Proceedings ISSAC'95, pp. 303-308. 1995.


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